WASHINGTON — President Donald Trump is imagining steel mill openings that aren’t happening and in denial about Russia’s role in the 2016 presidential election.

He points broadly to a “Russian hoax,” even as his own top national security and intelligence officials decry a real threat of Russian interference in future U.S. elections, and suggests that Russia didn’t want him to win the presidency. On jobs, he isn’t providing the full picture regarding U.S. employment growth.

The statements came in a week of puzzling disconnects and untruths, including assertions of cleaner air than his government recorded and a claim that people need photo IDs to make purchases in stores that any average shopper with cash or a credit card would know is not accurate.

A sampling of comments by Trump and his officials:

JOBS

TRUMP: “Here’s one of the best statistics ever. More Americans are now employed than ever recorded before in our nation’s history. Think of that. So we now have more people employed working today than the United States has ever had.” — Ohio rally Saturday.

THE FACTS: He’s painting an incomplete picture of U.S. job growth. Due largely to population increase, the number of people with jobs is, in fact, at a record high of 156 million. But a more relevant measure — the proportion of Americans with jobs — isn’t even close to a record.

Last month, 60.5 percent of Americans 16 and older had jobs. That is up from the recession and its aftermath, when many Americans stopped looking for work. It bottomed out at 58.2 percent in July 2011. Both figures are far below the record high of 64.7 percent, which was briefly reached in 2000. At the beginning of the 2008-2009 recession, 62.7 percent of Americans had jobs.

TRUMP: “U.S. Steel is opening up seven plants.” — remarks Thursday at the Pennsylvania rally. On Tuesday: “Thanks to our tariffs, our steel workers are back on the job, American steel mills are back open for business … U.S. Steel just announced that they’re building six new steel mills.” — Florida rally.

THE FACTS: No, U.S. Steel has not announced six, or seven, new steel mills. A spokeswoman for the Pittsburgh-based company, Meghan Cox, declined to comment on Trump’s claim, only making clear that any “operational changes” such as the opening of new mills would be “publicly announced” and “made available on our website” if it occurred.

RUSSIA INVESTIGATION

TRUMP: “I’ll tell you what, Russia’s very unhappy that Trump won, that I can tell you.” — Pennsylvania rally Thursday.

THE FACTS: That’s not what Russian President Vladimir Putin says. Asked at a news conference with Trump last month whether he wanted Trump to win the 2016 election, Putin responded, “Yes, I did.” Putin said he favored Trump “because he talked about bringing the U.S.-Russia relationship back to normal.”

The Republican-led Senate intelligence committee in May said it agreed with the U.S. intelligence agencies’ assessment that Russia intervened in the 2016 election to hurt the candidacy of Democrat Hillary Clinton and help Trump.

TRUMP: “In Helsinki, I had a great meeting with Putin. We discussed everything — I had a great meeting. We got along really well. By the way, that’s a good thing, not a bad thing. That’s a really good thing. Now we’re being hindered by the Russian hoax. It’s a hoax, OK?” — Pennsylvania rally.

THE FACTS: Trump’s effort to play down the idea of Russian election interference belies the statements of his own top national security and intelligence officials, who have concluded that Russia meddled in 2016 and pointed to a threat of future attacks on U.S. democracy. Just hours before Trump’s claim, National Intelligence Director Dan Coats joined other top aides at the White House to stress the Russia threat to U.S. elections “is real, it is continuing.”

National security adviser John Bolton said stemming the threat of Russian election interference is a priority for Trump, adding that the president had opened his private meeting with Putin by raising the issue. “I think the president has made it abundantly clear to everybody who has responsibility in this area that he cares deeply about it and that he expects them to do their jobs to their fullest ability and that he supports them fully,” he said.

On Sunday, White House officials sought to deny a disconnect between their message of a Russian meddling threat and Trump’s claims of a “hoax,” saying Trump was referring to federal investigations into possible collaboration between the Trump campaign and Russia.

But in recent weeks, special counsel Robert Mueller’s probe has led to indictments against 12 Russian military intelligence officers who allegedly sought to influence the U.S. election, the first time Moscow had been directly implicated in 2016 meddling. Congressional committees also have been investigating the matter, with the Senate intelligence committee making clear that a key goal is to improve security in future U.S. elections.

TRUMP: “Collusion is not a crime, but that doesn’t matter because there was No Collusion (except by Crooked Hillary and the Democrats)!” — tweet Tuesday.

TRUMP LAWYER RUDY GIULIANI: “I have been sitting here looking in the federal code trying to find collusion as a crime. … Collusion is not a crime.” — remarks July 30 on Fox News.

THE FACTS: It is correct to say election collusion isn’t a precise legal term. The U.S. code mostly uses the term “collusion” in antitrust laws to address crimes like price fixing. As it relates to Russia and U.S. elections, the term can be seen as shorthand for plenty of violations of specific laws on the books.

For instance, there could be legal violations if Trump’s presidential campaign is found to have collaborated with Moscow, including a conspiracy to defraud the United States. There are also laws against election fraud, computer hacking, wire fraud and falsifying records, if those apply.

So far, Mueller has accused the Russians of hacking into Democrats’ computers and stealing emails, as well as trying to stoke U.S. tensions before the 2016 election using social media.

Mueller might decide, for example, that a crime was committed if he finds evidence that an American was involved in the hack of Democrats, either by soliciting it or paying someone to do it.

As well, a conspiracy to defraud the United States can be used to refer to any two people using “deceit, craft, or trickery” to interfere with governmental functions, such as an election.

CLEAN AIR

TRUMP, on air quality in the U.S.: “It’s the best it’s ever been.” — Pennsylvania rally Thursday.

THE FACTS: Not true, going by the key measure of air quality. The EPA’s air quality index shows a worsening since 2014, the best year as measured by the number of days with bad air.

For that index, 35 cities reported unhealthy air for a total of 599 days in 2014. That went up to 729 days in 2017, the worst year since 2012 (1,297 days). The index measures ozone and soot.

AUTO SAFETY

EPA, citing potential benefits from freezing Obama-era mileage standards: “Increased vehicle affordability leading to increased driving of newer, safer, more efficient and cleaner vehicles. … Over 12,000 fewer crash fatalities over the lifetimes of all vehicles built through model year 2029. Up to 1,000 lives saved annually.” — information sheet released Thursday.

THE FACTS: The claimed safety benefits are unverifiable and probably overstated.

While newer vehicles are safer due to better engineering and safety features such as more air bags, automatic emergency braking and blind spot detection, auto safety experts say the difference between vehicles made 10 years ago and now isn’t that big and the number of lives saved can’t really be calculated.

Decade-old vehicles have anti-lock brakes and electronic stability control that stop drivers from losing control, two major safety advances.

THE FACTS: Insulating U.S. manufacturers is not easy to do. Even if the U.S. freezes its mileage requirements, the European Union, China, Japan and other nations will continue to increase theirs, which already are more stringent. Because most automakers sell vehicles worldwide, they’ll have to develop new technology such as electric cars anyway to satisfy other markets. The U.S. may not get the new technology as quickly as elsewhere.

VOTER ID

TRUMP: “We believe that only American citizens should vote in American elections, which is why the time has come for voter ID like everything else. If you go out and you want to buy groceries, you need a picture on a card, you need ID. You go out, you want to buy anything, you need ID, you need your picture.” — remarks Tuesday.

THE FACTS: As shoppers know, no photo is required to purchase items at retail stores with cash or to make routine purchases with credit or debit cards.

Identifications are required to purchase limited items such as alcohol, cigarettes or cold medicine and in rapidly declining situations in which a customer opts to pay with a personal check.

According to the National Grocers Association’s most recent data, the use of checks as a percentage of total transactions dropped from 33 percent in 2000 to 6 percent in 2015, due in part to the popularity of debit cards, which use PIN codes. The group’s members are independent food retailers, family-owned or privately held, both large and small.

WHITE HOUSE PRESS SECRETARY SARAH HUCKABEE SANDERS: “He’s not saying every time he went in; he said when you go to the grocery store.” — press briefing Wednesday.

THE FACTS: Actually, Trump did claim, erroneously, that photo IDs are required whenever “you want to buy anything,” not only in limited cases.

NATO

TRUMP: “I went to NATO. And NATO was essentially going out of business ’cause people weren’t paying and it was going down, down, down.” On NATO Secretary General Jens Stoltenberg: “He said we couldn’t collect money until President Trump came along. And he said last year we collected $44 billion. And this year the money is pouring in. … So the bottom line is the NATO countries are now paying a lot more money.” — news conference July 30 with Italian Prime Minister Giuseppe Conte.

THE FACTS: Countries don’t pay to be in NATO and don’t owe the organization anything other than contributions to a largely administrative fund that Trump is not talking about. Member countries are not in debt to NATO. Money is “not pouring in” now. Collections have not increased, as he asserted.

Trump’s actual beef is with how much NATO countries spend on their own military budgets.

The Trump administration is not the first to push countries in NATO to spend more on their own armed forces to lessen their dependence on the U.S. In fact, it was in 2014, during the Obama administration, that NATO members agreed to move “toward” spending 2 percent of their gross domestic product on their own defense by 2024.

The somewhat-vague commitment was made as a response to Russia’s actions in Ukraine and its annexation of Crimea. No one expected all allies would immediately move to 2 percent; the increases were to be gradual.

VETERANS

TRUMP: “We passed the biggest VA reform in half a century, Veteran’s Choice. If our veterans can’t get the care they need from the VA, they will have the right to go see a private doctor.” — remarks Tuesday in Tampa.

THE FACTS: Trump’s suggestion that veterans can get care immediately under the private-sector Veterans Choice program and without restriction is misleading.

Before veterans have the right to see a private doctor, they must meet certain criteria first, such as whether they face an “excessive burden” in receiving care at a Department of Veterans Affairs medical center. Under the current Choice guidelines, veterans also must wait at least 30 days for an appointment at a VA facility before they are eligible to receive care from a private doctor.

TRUMP: “And I used to say before I really was well-versed on the veteran situation in health care, I used to say all the time, ‘Why don’t they just let the folks go to a doctor?’ They’d wait in line for 7 days, 9 days, 14 days, 21 days … I said, ‘Why don’t they just let them go see a local private doctor, pay the bill and take care of it?’ And it’s turned out to be something that the veterans love. And it’s passed. It’s passed.” — remarks Tuesday.

THE FACTS: It’s not clear if veterans love the current Choice program, judging by the wait times. Despite the Choice program’s guarantee of providing appointments within 30 days, a recent report from the Government Accountability Office found that veterans actually waited an average of 51 to 64 days to receive care. A newly expanded Choice program will take at least a year to be implemented.